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In mathematics the estimation lemma, also known as the ML inequality, gives an upper bound for a contour integral. If f is a complex -valued, continuous function on the contour Γ and if its absolute value | f ( z ) | is bounded by a constant M for all z on Γ , then
Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount Bhatia–Davis inequality , an upper bound on the variance of any bounded probability distribution
The second inequality is the elementary inequality between and . The last inequality follows by applying reverse Fatou lemma , i.e. applying the Fatou lemma to the non-negative functions g − f n {\displaystyle g-f_{n}} , and again (up to sign) cancelling the finite ∫ X g d μ {\displaystyle \int _{X}g\,d\mu } term.
The standard absolute value on the integers. The standard absolute value on the complex numbers.; The p-adic absolute value on the rational numbers.; If R is the field of rational functions over a field F and () is a fixed irreducible polynomial over F, then the following defines an absolute value on R: for () in R define | | to be , where () = () and ((), ()) = = ((), ()).
The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist. The subdifferential of | x | at x = 0 is the interval [−1, 1]. [14] The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann ...
The feasible regions of linear programming are defined by a set of inequalities. In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. [1] It is used most often to compare two numbers on the number line by their size.
Grönwall's inequality is an important tool to obtain various estimates in the theory of ordinary and stochastic differential equations. In particular, it provides a comparison theorem that can be used to prove uniqueness of a solution to the initial value problem; see the Picard–Lindelöf theorem. It is named for Thomas Hakon Grönwall (1877 ...
Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality In mathematics , Jensen's inequality , named after the Danish mathematician Johan Jensen , relates the value of a convex function of an integral to the integral of the convex function.